The smallest squares containing k 58's :
15876 = 126²,   45805824 = 6768²,   1358585881 = 36859²,
15858585856656 = 3982284²,   16915815858585856 = 130060816².

The 2nd integer which is the sum of 4 squares in just 4 ways.
  [1,2,2,7], [1,4,4,5], [2,2,5,5], [2,3,3,6]
The 2nd integer which is the sum of 7 squares in just 7 ways.

58² = 3364, 3 * 3 * 6 + 4 = 58.

Loop of length 8 by the function f(N) = ... + c² + b² + a² for N = ... + 10²c + 10b + a:
58 -- 89 -- 145 -- 42 -- 20 -- 4 -- 16 -- 37 -- 58

58² is the first square which is the sum of 5 distinct fourth powers : [1,3,4,5,7].

58² = 3364, 3 + 3 + 6 + 4 = 4².

58² = (1² + 1)(41² + 1).

58⁸ = 128063081718016, 1 + 2 + 80 + 6 + 3081 + 7 + 180 + 1 + 6 = 58².

10^k + 44^k + 57^k + 58^k are squares for k = 1,2,3 (13², 93², 683²).
22^k + 24^k + 58^k + 65^k are squares for k = 1,2,3 (13², 93², 703²).
14^k + 58^k + 122^k + 130^k are squares for k = 1,2,3 (18², 188², 2052²).
58^k + 7018^k + 10034^k + 13166^k are squares for k = 1,2,3 (174², 17980², 1907388²).

Komachi Fraction : (58/103)² = 30276/95481.

Komachi equations:
58² = 1 * 2 * 34 * 5 + 6 * 7 * 8 * 9 = 1 - 2 - 34 + 5 * 678 + 9
  = - 1 / 2 * 34 + 5 * 678 - 9 = 9 * 87 + 6 * 5 * 43 * 2 + 1
  = 98 + 7 + 6 * 543 + 2 - 1 = - 9 - 8 + 765 * 4 + 321
  = - 987 + 6 * 5 + 4321 = 9 * 8 * 7 * 6 + 5 * 4 + 32 * 10
  = 98 + 76 - 5 * 4 + 3210,
58² = - 12² * 3² / 4² + 5² - 6² * 7² + 8² * 9² = 9² + 8² + 7² - 65² + 43² * 2² - 1²
  = - 98² / 7² - 6² + 5² * 4² * 3² - 2² */ 1² = 9² + 8² * 7² + 6² - 5² + 4² * 3² / 2² + 10²
  = 98² - 76² - 5² * 4² + 3² * 2² - 10² = - 9² - 8² + 7² - 6² + 5² * 4² * 3² - 2² - 10²
  = - 9² + 8² * 7² + 6² + 5² + 4² * 3² + 2² + 10²,
58² = 98³ / 7³ + 6³ + 5³ + 4³ + 3³ * 2³ - 1³.

(58² + 2) = (1² + 2)(3² + 2)(10² + 2) = (2² + 2)(3² + 2)(7² + 2) = (7² + 2)(8² + 2),
(58² - 4) = (3² - 4)(26² - 4) = (3² - 4)(5² - 4)(6² - 4).

58² + 59² + 60² + ... + 2132² = 56855².

(1)(2 + 3 + ... + 55)(56 + 57 + 58) = 513²,
(1 + 2)(3 + 4)(5 + 6 + ... + 58) = 189²,
(1 + 2)(3 + 4 + ... + 32)(33 + 34 + ... + 58) = 1365²,
(1 + 2 + 3)(4 + 5 + 6)(7 + 8 + ... + 58) = 390²,
(1 + 2 + 3)(4 + 5 + ... + 31)(32 + 33 + ... + 58) = 1890²,
(1 + 2 + 3 + 4)(5 + 6 + ... + 31)(32 + 33 + ... + 58) = 2430²,
(1 + 2 + 3 + 4)(5 + 6 + ... + 49)(50 + 51 + ... + 58) = 2430²,
(1 + 2 + 3 + 4 + 5)(6)(7 + 8 + ... + 58) = 390²,
(1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 49)(50 + 51 + ... + 58) = 2970²,
(1 + 2 + 3 + 4 + 5 + 6)(7 + 8 + ... + 32)(33 + 34 + ... + 58) = 3549²,
(1 + 2 + ... + 11)(12 + 13 + ... + 32)(33 + 34 + ... + 58) = 6006²,
(1 + 2 + ... + 14)(15 + 16 + ... + 49)(50 + 51 + ... + 58) = 7560²,
(1 + 2 + ... + 15)(16 + 17 + ... + 21)(22 + 23 + ... + 58) = 4440²,
(1 + 2 + ... + 16)(17 + 18 + ... + 43)(44 + 45 + ... + 58) = 9180²,
(1 + 2 + ... + 18)(19)(20 + 21 + ... + 58) = 2223²,
(1 + 2 + ... + 18)(19 + 20 + ... + 36)(37 + 38 + ... + 58) = 9405²,
(1 + 2 + ... + 18)(19 + 20 + ... + 37)(38 + 39 + ... + 58) = 9576²,
(1 + 2 + ... + 18)(19 + 20 + ... + 55)(56 + 57 + 58) = 6327²,
(1 + 2 + ... + 21)(22 + 23 + ... + 32)(33 + 34 + ... + 58) = 9009²,
(1 + 2 + ... + 27)(28 + 29 + ... + 32)(33 + 34 + ... + 58) = 8190²,
(1 + 2 + ... + 27)(28 + 29 + ... + 49)(50 + 51 + ... + 58) = 12474²,
(1 + 2 + ... + 30)(31)(32 + 33 + ... + 58) = 4185²,
(1 + 2 + ... + 38)(39)(40 + 41 + ... + 58) = 5187²,
(1 + 2 + ... + 48)(49)(50 + 51 + ... + 58) = 5292².

58² = 3364 appears in the decimal expressions of π and e:
  π = 3.14159•••3364••• (from the 6247th digit),
  e = 2.71828•••3364••• (from the 7446st digit).

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Page of Squares : First Upload January 26, 2004 ; Last Revised November 2, 2013
by Yoshio Mimura, Kobe, Japan